Spectral synthesis with the complexity parameter

We show that spectral synthesis thresholds are governed by a quantitative spectral complexity parameter, the Fourier Ratio, in addition to the geometric size of the Fourier support. In the Euclidean setting, we prove that if a compactly supported measure has finite $\alpha$-dimensional packing measure and the associated Fourier ratio decays with asymptotic exponent $\kappa$, then the classical synthesis threshold improves from $\frac{2d}{\alpha}$ to $\frac{2(d-2\kappa)}{\alpha-2\kappa}$. We then establish an analogous result on compact Riemannian manifolds without boundary. In that setting the relevant object is a localized spectral Fourier ratio defined using Laplace--Beltrami spectral projectors. The resulting synthesis threshold is again determined by the decay exponent of this complexity parameter. These results place Euclidean and manifold spectral synthesis into a common framework in which geometric size and spectral complexity jointly govern uniqueness